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Low Rank and Sparse Decompositionof Ultrasound Color Flow Images

for Suppressing Clutter in Real-TimeMd Ashikuzzaman, Clyde Belasso, Md. Golam Kibria, Andreas Bergdahl, Claudine J. Gauthier and Hassan Rivaz

In this supplementary material, we report a schematic di-agram of the set-up for the phantom experiment (Fig. 1). Inaddition, we represent an analysis to show the dependenceof the optimal tunable parameters of SVD on the noiselevel (Fig. 2). Finally, we present a comparison among thepower Doppler images obtained from SVD, HOSVD [1] andRAPID (Fig. 3).

I. RESULTS

Fig. 1 shows the schematic description of the set-up for thephantom experiment.

Fig. 2 represents the power Doppler images from SVD andRAPID for the simulation data with added random noise ofuniform distribution. Two levels of noise with Peak Signal-to-Noise Ratio (PSNR) values of 58.43 dB and 39.34 dB areadded to the envelopes of RF data. It is evident from Fig. 2that the result from SVD for blood rank 15 is similar to thatof 19 in case of the lower noise level. On that other hand,the results from SVD for blood subspace ranks 15 and 19 aresubstantially different from each other for the higher level ofnoise. This study indicates that the optimal values of tunableparameters of SVD are highly dependent on the noise level. Onthe contrary, RAPID automatically obtains the optimal resultregardless of the level of noise.

Fig. 3 depicts the clutter suppressed power Doppler imagesfor simulation, phantom and in-vivo data sets generated bySVD, HOSVD and RAPID. We have incorporated 15 Radio-Frequency (RF) frames to generate the power Doppler imagesfrom SVD and RAPID. We consider a data tensor consistingof 3 matrices where each matrix is an ensemble of 15 slowtime frames. For SVD and HOSVD, the best results obtainedby careful tuning of the parameters are reported. RAPIDconverges to the optimal results without the need of anymanual tuning. The results from SVD and RAPID are verysimilar to each other. HOSVD does not seem to improve thequality of the power Doppler images for the datasets usedin this study. However, HOSVD is expected to improve theresult when a large number of data matrices consisting of moreslow time frames are incorporated to form the data tensor.Therefore, it is suggested that HOSVD improves the result atthe expense of extensive amount of data. Besides, this methodsuffers from much higher running time than SVD and RAPID.To be precise, our MATLAB implementation of HOSVD takesmore than 40 minutes to execute for a tensor of 3 matrices eachof which consists of 15 slow time frames of size 250×125. On

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Fig. 1: A schematic depiction of the set-up for the phantomexperiment.

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Fig. 2: Power Doppler images for simulation with differentnoise levels. Rows 1 and 2 correspond to PSNR values of58.43 dB and 39.34 dB respectively. Columns 1 and 2 showresults from SVD for different combinations of subspaceranks. Column 3 represents the results from RAPID. (g) showsthe color bar.

the other hand, both SVD and RAPID take less than 1 secondto process the same amount of data. Another limitation ofHOSVD is that it has 6 tunable parameters and there is norigid criterion to select the optimal set of values for them.Hence it is very difficult to obtain the optimal result whiledealing with a large dataset since it is subject to the manualtuning of 6 parameters over a certain range. This drawbackcalls the clinical usefulness of HOSVD into question.

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Fig. 3: Power Doppler images for simulation, phantom and in-vivo data sets. Rows 1-5 correspond to simulation, phantomwith focused imaging, phantom with plane wave imaging, in-vivo data from a rat’s abdomen and in-vivo data from the knee ofa human subject respectively. Columns 1-4 depict B-mode, power Doppler images obtained from SVD, HOSVD and RAPIDrespectively. (u), (v), (w), (x) and (y) represent the color bars for power Doppler images obtained from simulation, phantomwith focused imaging, phantom with plane wave imaging, in-vivo data from a rat’s abdomen and in-vivo data from the kneeof a human subject respectively.

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REFERENCES

[1] M. Kim, C. K. Abbey, J. Hedhli, L. W. Dobrucki, and M. F. Insana,“Expanding acquisition and clutter filter dimensions for improved perfu-

sion sensitivity,” IEEE Transactions on Ultrasonics, Ferroelectrics, andFrequency Control, vol. 64, no. 10, pp. 1429–1438, 2017.